Let P{\displaystyle P} be a set of program clauses. Assume a selection function, which gives for a given goal ←A1,…,An{\displaystyle \gets A_{1},\ldots ,A_{n}} one of its subgoals Ai{\displaystyle A_{i}}. Further assume a goal Gi=←A1,…,Am,…,An{\displaystyle G_{i}=\gets A_{1},\ldots ,A_{m},\ldots ,A_{n}} and a selection function which selects Am{\displaystyle A_{m}}. Let Ci=A←B1,…,Bq{\displaystyle C_{i}=A\gets B_{1},\ldots ,B_{q}} be a variant of a clause in P{\displaystyle P}, such that Ci{\displaystyle C_{i}} and Gi{\displaystyle G_{i}} have no variable in common. If θi+1{\displaystyle \theta _{i+1}} is most general unifier of Am{\displaystyle A_{m}} and A{\displaystyle A}, the goal

An SLD-deduction (-refutation)of P∪{G}{\displaystyle P\cup \{G\}} for a set of program clauses P{\displaystyle P} and a goal clause G{\displaystyle G} is a linear deduction (refutation) in which only SLD-resolution steps occur and G{\displaystyle G} is the start clause.

Let P{\displaystyle P} be a set of program clauses, G{\displaystyle G} a goal clause and R{\displaystyle R} a selection function. For every correct answer substitution θ{\displaystyle \theta } for P∪{G}{\displaystyle P\cup \{G\}}, there is an R{\displaystyle R}-computed answer substitution σ{\displaystyle \sigma } for P∪{G}{\displaystyle P\cup \{G\}} and a substitution γ{\displaystyle \gamma }, such that θ=σ∘γ|Var(G){\displaystyle \theta =\sigma \circ \gamma \;|_{Var(G)}}

In the Section on Propositional Logic we already explained propositional tableaux and its variants, like the connection calculus and model elimination. In this section we will give model elimination in the first order case. Note that we need one more inference rule, the reduction rule, in this case

A clause (normalform) tableau for a set of clauses S{\displaystyle S} is a tableau for S{\displaystyle S}, whose nodes are literals from S{\displaystyle S} and which is constructed by a (possibly infinite) sequence of applications of the following rules:

The tree consisting of root true{\displaystyle true} and immediate successors L1,⋯,Ln{\displaystyle L_{1},\cdots ,L_{n}}, where C=L1,⋯,Ln{\displaystyle C=L_{1},\cdots ,L_{n}} is a new variant of a clause from S{\displaystyle S} is a tableau for S{\displaystyle S} (initialisation rule).

Let T{\displaystyle T} be a tableau for S{\displaystyle S}, B{\displaystyle B} a branch of T{\displaystyle T}, and C=L1,⋯,Ln{\displaystyle C=L_{1},\cdots ,L_{n}} an new variant of a clause from S{\displaystyle S}, such that the link-condition with mgu σ{\displaystyle \sigma } is satisfied. If the tree T′{\displaystyle T'} is constructed by extending B{\displaystyle B} by the n{\displaystyle n} subtrees Li{\displaystyle L_{i}}, then T′σ{\displaystyle T'\sigma } is a tableau for S{\displaystyle S} (expansion rule).

Let T{\displaystyle T} be a tableau for S{\displaystyle S}, B{\displaystyle B} a branch of T{\displaystyle T}, L{\displaystyle L} a leaf of B{\displaystyle B}, and L′∈C{\displaystyle L'\in C}, such that L′¯{\displaystyle {\overline {L'}}} and L{\displaystyle L} have a mgu σ{\displaystyle \sigma }, than Tσ{\displaystyle T\sigma } is a tableau for S{\displaystyle S} (reduction rule).

The following are three possible link conditions:

No condition.

Weak link condition: There is a literal L∈B{\displaystyle L\in B} and L′∈C{\displaystyle L'\in C}, such that L′¯{\displaystyle {\overline {L'}}} and L{\displaystyle L} have a mgu σ{\displaystyle \sigma }

Strong link condition: There is a leaf L{\displaystyle L} of B{\displaystyle B}, and L′∈C{\displaystyle L'\in C}, such that L′¯{\displaystyle {\overline {L'}}} and L{\displaystyle L} have a mgu σ{\displaystyle \sigma } .

Analog to the propositional case the different link conditions result in different calculi: